Contents

Jan 19

Bianca Viray

On the dependence of the Brauer-Manin obstruction on the degree of a variety

Let X be a smooth projective variety of degree d over a number field k. In 1970 Manin observed that elements of the Brauer group of X can obstruct the existence of a k-point, even when X is everywhere locally soluble. In joint work with Brendan Creutz, we prove that if X is geometrically abelian, Kummer, or bielliptic then this Brauer-Manin obstruction to the existence of a k-point can be detected from only the d-primary torsion Brauer classes.

Feb 2

Given a number field K of fixed degree n over Q, a classical theorem of Brauer--Siegel asserts that the size of the class group of K is bounded by O_\epsilon(|Disc(K)|^(1/2+\epsilon). For any prime p, it is conjectured that the p-torsion
subgroup of the class group of K is bounded by O_\epsilon(|Disc(K)|^\epsilon. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" Brauer--Siegel bound.

In this talk, we will discuss a proof of a subconvex bound on the size of the 2-torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the number of integral points on elliptic curves.

Feb 9

Tonghai Yang

L-function aspect of the Colmez Conjecture

Associate to a CM type, Colmez defined two invariants: Faltings height of the associated CM abelian varieties of this CM type, and the log derivative of some mysterious Artin L-function nifonstructed from this CM type. Furthermore, he conjectured them to be equal and proved the conjecture for Abelian CM number fields (up to log 2). The average version of the conjecture was proved recently by two groups of people which has significant implication to Andre-Oort conjecture. Some non-abelian cases were proved by myself and others. In all proved cases, the L-function is either Dirichlet characters or quadratic Hecke characters. A natural question is what kinds of Artin L-functions show up in this conjecture. In this talk, we will talk about some interesting examples in this. This is a joint work with Hongbo Yin.

Feb 16

Alexandra Florea

Moments of L-functions over function fields

I will talk about the moments of the family of quadratic Dirichlet L–functions over function fields. Fixing the finite field and letting the genus of the family go to infinity, I will explain how to obtain asymptotic formulas for the first four moments in the hyperelliptic ensemble.

Feb 23

Dongxi Ye

Borcherds Products on Unitary Group U(2,1)

In this talk, I will first briefly go over the concepts of Borcherds products on orthogonal groups and unitary groups. And then I will present a family of new explicit examples of Borcherds products on unitary group U(2,1), which arise from a canonical basis for the space of weakly holomorphic modular forms of weight $-1$ for $\Gamma_{0}(4)$. This talk is based on joint work with Professor Tonghai Yang.

Mar 2

Frank Thorne

Levels of distribution for prehomogeneous vector spaces

One important technical ingredient in many arithmetic statistics papers is

upper bounds for finite exponential sums which arise as Fourier transforms
of characteristic functions of orbits. This is typical in results
obtaining power saving error terms, treating "local conditions", and/or
applying any sort of sieve.

In my talk I will explain what these exponential sums are, how they arise,
and what their relevance is. I will outline a new method for explicitly and easily
evaluating them, and describe some pleasant surprises in our end results. I will also
outline a new sieve method for efficiently exploiting these results, involving
Poisson summation and the Bhargava-Ekedahl geometric sieve. For example, we have proved
that there are "many" quartic field discriminants with at most eight
prime factors.